Undirected Graphs...Planarity. Can you draw the graph in the plane with no crossing edges? Graph isomorphism. Do two adjacency matrices represent the same graph? Challenge. Which of

Algorithms in Java, 4th Edition · Robert Sedgewick and Kevin Wayne · Copyright © 2008 · October 21, 2008 1:36:19 AM

Undirected Graphs

References:

Algorithms in Java, Chapters 17 and 18 http://www.cs.princeton.edu/algs4/51undirected

! graph API

! maze exploration

! depth-first search

! breadth-first search

! connected components

! challenges

Graph. Set of vertices connected pairwise by edges.

Why study graph algorithms?

• Interesting and broadly useful abstraction.

• Challenging branch of computer science and discrete math.

• Hundreds of graph algorithms known.

• Thousands of practical applications.

2

Undirected graphs

3

Protein interaction network

Reference: Jeong et al, Nature Review | Genetics

4

The Internet as mapped by the Opte Project

http://en.wikipedia.org/wiki/Internet

5

Social networks

Reference: Bearman, Moody and Stovel, 2004image by Mark Newman

high school dating

Reference: Adamic and Adar, 2004

corporate e-mail

7

Graph applications

graph vertex edge

communication telephone, computer fiber optic cable

circuit gate, register, processor wire

mechanical joint rod, beam, spring

financial stock, currency transactions

transportation street intersection, airport highway, airway route

internet class C network connection

game board position legal move

social relationship person, actor friendship, movie cast

neural network neuron synapse

protein network protein protein-protein interaction

chemical compound molecule bond

8

Graph terminology

9

Some graph-processing problems

Path. Is there a path between s and t?

Shortest path. What is the shortest path between s and t?

Cycle. Is there a cycle in the graph?

Euler tour. Is there a cycle that uses each edge exactly once?

Hamilton tour. Is there a cycle that uses each vertex exactly once?

Connectivity. Is there a way to connect all of the vertices?

MST. What is the best way to connect all of the vertices?

Biconnectivity. Is there a vertex whose removal disconnects the graph?

Planarity. Can you draw the graph in the plane with no crossing edges?

Graph isomorphism. Do two adjacency matrices represent the same graph?

Challenge. Which of these problems are easy? difficult? intractable?

10

! graph API

! maze exploration




! challenges

Vertex representation.

• This lecture: use integers between 0 and V-1.

• Real world: convert between names and integers with symbol table.

Other issues. Parallel edges, self-loops.

A

G

E

CB

F

D

11

Graph representation

symbol table

0

6

4

21

5

3

12

Graph API

public class Graph graph data type

Graph(int V) create an empty graph with V vertices

Graph(In in) create a graph from input stream

void addEdge(int v, int w) add an edge v-w

Iterable<Integer> adj(int v) return an iterator over the neighbors of v

int V() return number of vertices

String toString() return a string representation

In in = new In();

Graph G = new Graph(in);

StdOut.println(G);

for (int v = 0; v < G.V(); v++)

for (int w : G.adj(v))

/* process edge v-w */

read graph from

standard input

process both

v-w and w-v

% more tiny.txt

7

0 1

0 2

0 5

0 6

3 4

3 5

4 6

Store a list of the edges (linked list or array).

13

Set of edges representation

87

109

1211

0

6

4

21

5

3

0 1

0 2

0 5

0 6

3 4

3 5

4 6

7 8

9 10

9 11

9 12

Maintain a two-dimensional V-by-V boolean array;

for each edge v-w in graph: adj[v][w] = adj[w][v] = true.

0 1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

9

10

11

12

0 1 1 0 0 1 1 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0 0 0

0 0 0 1 0 1 0 0 0 0 0 0 0

1 0 0 1 1 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 1 1

0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 1

0 0 0 0 0 0 0 0 0 1 0 1 0

14

Adjacency-matrix representation

two entries

for each edge

87

109

1211

0

6

4

21

5

3

15

Adjacency-matrix representation: Java implementation

public class Graph

{

private final int V;

private final boolean[][] adj;

public Graph(int V)

{

this.V = V;

adj = new boolean[V][V];

}

public void addEdge(int v, int w)

{

adj[v][w] = true;

adj[w][v] = true;

}

public Iterable<Integer> adj(int v)

{ return new AdjIterator(v); }

}

adjacency matrix

create empty graph

with V vertices

add edge v-w

(no parallel edges)

iterator for v's neighbors

(code for AdjIterator omitted)

16

Adjacency-list representation

Maintain vertex-indexed array of lists (implementation omitted).

5 2 1 6

0

0

5 4

6 5 3

0 4 3

4 0

8

7

10 11 12

9

9 12

9 11

two entries

for each edge

0:

1:

2:

3:

4:

5:

6:

7:

8:

9:

10:

11:

12:

87

109

1211

0

6

4

21

5

3

Maintain vertex-indexed array of sets.

0:

1:

2:

3:

4:

5:

6:

7:

8:

9:

10:

11:

12:

{ 1 2 5 6 }

{ 0 }

{ 0 }

{ 4, 5 }

{ 3, 5, 6 }

{ 0, 3, 4 }

{ 0, 4 }

{ 8 }

{ 7 }

{ 10, 11, 12 }

{ 9 }

{ 9, 12 }

{ 1, 9 }

17

Adjacency-set graph representation

two entries

for each edge

87

109

1211

0

6

4

21

5

3

18

Adjacency-set representation: Java implementation

public class Graph

{

private final int V;

private final SET<Integer>[] adj;

public Graph(int V)

{

this.V = V;

adj = (SET<Integer>[]) new SET[V];

for (int v = 0; v < V; v++)

adj[v] = new SET<Integer>();

}

public void addEdge(int v, int w)

{

adj[v].add(w);

adj[w].add(v);

}

public Iterable<Integer> adj(int v)

{ return adj[v]; }

}

adjacency sets

create empty graph

with V vertices

add edge v-w

(no parallel edges)

iterator for v's neighbors

In practice. Use adjacency-set (or adjacency-list) representation.

• Algs all based on iterating over edges incident to v.

• Real-world graphs tend to be “sparse.”

19

Graph representations

representation spaceedge between

v and w?iterate over edges

incident to v?

list of edges E E E

adjacency matrix V2 1 V

adjacency list E + V degree(v) degree(v)

adjacency set E + V log (degree(v)) degree(v)

huge number of vertices,

small average vertex degree

20

! graph API

! maze exploration




! challenges

21

Maze exploration

Maze graphs.

• Vertex = intersection.

• Edge = passage.

Goal. Explore every passage in the maze.

22

Trémaux maze exploration

Algorithm.

• Unroll a ball of string behind you.

• Mark each visited intersection by turning on a light.

• Mark each visited passage by opening a door.

First use? Theseus entered labyrinth to kill the monstrous Minotaur;

Ariadne held ball of string.

Claude Shannon (with Theseus mouse)

23

24

Maze exploration

25

Maze exploration

27

! graph API

! maze exploration




! challenges

Goal. Systematically search through a graph.

Idea. Mimic maze exploration.

Running time.

• O(E) since each edge examined at most twice.

• Usually less than V to find paths in real graphs.

• Typical applications.

• Find all vertices connected to a given s.

• Find a path from s to t.

Depth-first search

Mark s as visited.

Recursively visit all unmarked vertices v adjacent to s.

DFS (to visit a vertex s)

29

Design goal. Decouple graph data type from graph processing.

Typical client program.

• Create a Graph.

• Pass the Graph to a graph-processing routine, e.g., DFSearcher.

• Query the graph-processing routine for information.

Design pattern for graph processing

// print all vertices connected to s

In in = new In(args[0]);

Graph G = new Graph(in);

int s = 0;

DFSearcher dfs = new DFSearcher(G, s);

for (int v = 0; v < G.V(); v++)

if (dfs.isConnected(v))

StdOut.println(v);

30

Depth-first search (connectivity)

public class DFSearcher

{

private boolean[] marked;

public DFSearcher(Graph G, int s)

{

marked = new boolean[G.V()];

dfs(G, s);

}

private void dfs(Graph G, int v)

{

marked[v] = true;


if (!marked[w]) dfs(G, w);

}

public boolean isConnected(int v)

{ return marked[v]; }

}

true if connected to s

constructor marks

vertices connected to s

recursive DFS does the work

client can ask whether any

vertex is connected to s

Flood fill

Photoshop “magic wand”

31

Graph-processing challenge 1

Problem. Flood fill.

Assumptions. Picture has millions to billions of pixels.

How difficult?

• Any COS 126 student could do it.

• Need to be a typical diligent COS 226 student.

• Hire an expert.

• Intractable.

• No one knows.

• Impossible.

32

33

Connectivity application: flood fill

Change color of entire blob of neighboring red pixels to blue.

Build a grid graph.

• Vertex: pixel.

• Edge: between two adjacent red pixels.

• Blob: all pixels connected to given pixel.

recolor red blob to blue34

Connectivity application: flood fill

Change color of entire blob of neighboring red pixels to blue.

Build a grid graph.

• Vertex: pixel.

• Edge: between two adjacent red pixels.

• Blob: all pixels connected to given pixel.

recolor red blob to blue


Problem. Is there a path from s to t ?

How difficult?



• Hire an expert.

• Intractable.

• No one knows.

35

Problem. Find a path from s to t ?

Assumption. Any path will do.

How difficult?



• Hire an expert.

• Intractable.

• No one knows.


36

37

Paths in graphs: union find vs. DFS

Is there a path from s to t?

If so, find one.

• Union-find: not much help (run DFS on connected subgraph).

• DFS: easy (stay tuned).

Union-find advantage. Can intermix queries and edge insertions.

DFS advantage. Can recover path itself in time proportional to its length.

method preprocessing time query time space

union-find V + E log* V log* V † V

DFS E + V 1 E + V

† amortized

38

Keeping track of paths with DFS

DFS tree. Upon visiting a vertex v for the first time, remember that you

came from pred[v] (parent-link representation).

Retrace path. To find path between s and v, follow pred[] back from v.

39

Depth-first-search (pathfinding)

public class DFSearcher

{

private int[] pred;

...

public DFSearcher(Graph G, int s)

{

...

pred = new int[G.V()];

for (int v = 0; v < G.V(); v++)

pred[v] = -1;

...

}


{

marked[v] = true;


if (!marked[w])

{

pred[w] = v;

dfs(G, w);

}

}

public Iterable<Integer> path(int v)

{ /* see next slide */ }

}

add instance variable for parent-link

representation of DFS tree

initialize it in the constructor

set parent link

add method for client

to iterate through path

40

Depth-first-search (pathfinding iterator)

public Iterable<Integer> path(int v)

{

Stack<Integer> path = new Stack<Integer>();

while (v != -1 && marked[v])

{

list.push(v);

v = pred[v];

}

return path;

}

41

DFS summary

Enables direct solution of simple graph problems.

• Find path from s to t.

• Connected components (stay tuned).

• Euler tour (see book).

• Cycle detection (simple exercise).

• Bipartiteness checking (see book).

Basis for solving more difficult graph problems.

• Biconnected components (see book).

• Planarity testing (beyond scope).

!

42

! graph API

! maze exploration




! challenge

Depth-first search. Put unvisited vertices on a stack.

Breadth-first search. Put unvisited vertices on a queue.

Shortest path. Find path from s to t that uses fewest number of edges.

Property. BFS examines vertices in increasing distance from s.

43

Breadth-first search

Put s onto a FIFO queue.

Repeat until the queue is empty:

! remove the least recently added vertex v

! add each of v's unvisited neighbors to the queue,

and mark them as visited.

BFS (from source vertex s)

44

Breadth-first search scaffolding

public class BFSearcher

{

private int[] dist;

public BFSearcher(Graph G, int s)

{

dist = new int[G.V()];

for (int v = 0; v < G.V(); v++)

dist[v] = G.V() + 1;

dist[s] = 0;

bfs(G, s);

}

public int distance(int v)

{ return dist[v]; }

private void bfs(Graph G, int s)

{ /* See next slide */ }

}

initialize distances

distances from s

compute distances

answer client query

45

Breadth-first search (compute shortest-path distances)

private void bfs(Graph G, int s)

{

Queue<Integer> q = new Queue<Integer>();

q.enqueue(s);

while (!q.isEmpty())

{

int v = q.dequeue();


{

if (dist[w] > G.V())

{

q.enqueue(w);

dist[w] = dist[v] + 1;

}

}

}

}

46

BFS application

• Facebook.

• Kevin Bacon numbers.

• Fewest number of hops in a communication network.

ARPANET

47

! graph API

! maze exploration




! challenge

Def. Vertices v and w are connected if there is a path between them.

Def. A connected component is a maximal set of connected vertices.

Goal. Preprocess graph to answer queries: is v connected to w?

in constant time

Union-Find? Not quite.

48

Connectivity queries

Vertex Component

0 0

1 1

2 1

3 0

4 0

5 0

6 2

7 0

8 2

9 1

10 0

11 0

12 1

68

19

122

0

7

5

1011

3

3

Goal. Partition vertices into connected components.

49

Connected components

Initialize all vertices v as unmarked.

For each unmarked vertex v, run DFS and identify all vertices

discovered as part of the same connected component.


preprocess time query time extra space

E + V 1 V

50

Depth-first search for connected components

public class CCFinder

{

private final static int UNMARKED = -1;

private int components;

private int[] cc;

public CCFinder(Graph G)

{ /* see next slide */ }

public int connected(int v, int w)

{ return cc[v] == cc[w]; }

}

constant-time

connectivity query

component labels

51

Depth-first search for connected components

public CCFinder(Graph G)

{

cc = new int[G.V()];

for (int v = 0; v < G.V(); v++)

cc[v] = UNMARKED;

for (int v = 0; v < G.V(); v++)

if (cc[v] == UNMARKED)

{

dfs(G, v);

components++;

}

}


{

cc[v] = components;


if (cc[w] == UNMARKED) dfs(G, w);

}

DFS for each component

standard DFS

52


63 components

53

Connected components application: image processing

Goal. Read in a 2D color image and find regions of connected pixels

that have the same color.

Input. Scanned image.

Output. Number of red and blue states.

assuming contiguous states

Goal. Read in a 2D color image and find regions of connected pixels

that have the same color.

Efficient algorithm.

• Create grid graph.

• Connect each pixel to neighboring pixel if same color.

• Find connected components in resulting graph.

7 7

7 7

3

3

1 1 1 1 1

1

1

1 1

1 1

11

11 11

11 11 11 11

54

Connected components application: image processing

0 6 6

0 0 0 6 6 6 8

0 0 6 6 4 8

0 0 6 2 11

10 10 10 10 2 11

2 2 2 2 2 11

5 5 5 2 2 11

8 9 9

8 9

11 11 11 11

11 11 11 11

11 11 11 11

11 11 11 11

55

Connected components application: particle detection

Particle detection. Given grayscale image of particles, identify "blobs."

• Vertex: pixel.

• Edge: between two adjacent pixels with grayscale value ! 70.

• Blob: connected component of 20-30 pixels.

Particle tracking. Track moving particles over time.

black = 0

white = 255

56

! graph API

! maze exploration




! challenges


Problem. Find a cycle that uses every edge.

Assumption. Need to use each edge exactly once.

How difficult?



• Hire an expert.

• Intractable.

• No one knows.

• Impossible.

57

0-1

0-2

0-5

0-6

1-2

2-3

2-4

3-4

4-5

4-6

6

4

21

3

0

5

0-1-2-3-4-2-0-6-4-5-0

The Seven Bridges of Königsberg. [Leonhard Euler 1736]

Euler tour. Is there a cyclic path that uses each edge exactly once?

Answer. Yes iff connected and all vertices have even degree.

To find path. DFS-based algorithm (see Algs in Java).

58

Bridges of Königsberg

“ … in Königsberg in Prussia, there is an island A, called the

Kneiphof; the river which surrounds it is divided into two branches ...

and these branches are crossed by seven bridges. Concerning these

bridges, it was asked whether anyone could arrange a route in such a

way that he could cross each bridge once and only once. ”


Problem. Find a cycle that visits every vertex.

Assumption. Need to visit each vertex exactly once.

How difficult?



• Hire an expert.

• Intractable.

• No one knows.

• Impossible.

59

0-1

0-2

0-5

0-6

1-2

2-6

3-4

3-5

4-5

4-6

6

4

21

3

0

5

0-5-3-4-6-2-1-0


Problem. Are two graphs identical except for vertex names?

How difficult?



• Hire an expert.

• Intractable.

• No one knows.

• Impossible.

60

0-1

0-6

0-2

4-3

5-3

5-4

0-5

6-4

6

4

21

5

3

0

2-1

2-4

2-0

6-5

5-3

3-6

2-3

6-4

4

6

0

2

3

5

1


Problem. Lay out a graph in the plane without crossing edges?

How difficult?



• Hire an expert.

• Intractable.

• No one knows.

• Impossible.

61

0-2

1-2

2-3

2-4

3-5

3-6

4-6

5-6

4

6

0

2

3

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Documents

Undirected Graphs...Planarity. Can you draw the graph in the plane with no crossing edges? Graph isomorphism. Do two adjacency matrices represent the same graph? Challenge. Which of